134 ANALYTICAL MECHANICS velocity of the bullet, find the expression for the highest point reached. Also find the time of upward flight. 6. In the preceding problem suppose the resistance to be proportional to the velocity. 113. IV. Simple Harmonic Motion.—The motion of a particle is called simple harmonic when the particle is acted upon by a force which is always directed towards a fixed point and the magnitude of which is proportional to the distance of the particle from the same point. FIG. 71. Let m be the mass of the particle, the line A A' its path, Fig. 71, and 0 the fixed point. Then denoting the displacement, i.e., the distance of the particle from the fixed point, by x we obtain or dv 7 9 m— = — k2x, at for the force equation. It is evident that equation (1) is nothing more or less than the analytical expression for the foregoing definition of simple harmonic motion. The factor k2 is a constant. The negative sign in equation (1) indicates the fact that the force and the acceleration are directed towards the fixed point, while x is measured from